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# Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2018: 0.27

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ISSN
1572-9176
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Volume 26, Issue 2

# On Küneth’s correlation and its applications

Leonard Mdzinarishvili
Published Online: 2019-04-09 | DOI: https://doi.org/10.1515/gmj-2019-2021

## Abstract

Let $\mathcal{𝒦}$ be an abelian category that has enough injective objects, let $T:\mathcal{𝒦}\to A$ be any left exact covariant additive functor to an abelian category A and let ${T}^{\left(i\right)}$ be a right derived functor, $u\ge 1$, [S. Mardešić, Strong Shape and Homology, Springer Monogr. Math., Springer, Berlin, 2000]. If ${T}^{\left(i\right)}=0$ for $i\ge 2$ and ${T}^{\left(i\right)}{C}_{n}=0$ for all $n\in ℤ$, then there is an exact sequence

$0⟶{T}^{\left(1\right)}{H}_{n+1}\left({C}_{*}\right)⟶{H}_{n}\left(T{C}_{*}\right)⟶T{H}_{n}\left({C}_{*}\right)⟶0,$

where ${C}_{*}=\left\{{C}_{n}\right\}$ is a chain complex in the category $\mathcal{𝒦}$, ${H}_{n}\left({C}_{*}\right)$ is the homology of the chain complex ${C}_{*}$, $T{C}_{*}$ is a chain complex in the category A, and ${H}_{n}\left(T{C}_{*}\right)$ is the homology of the chain complex $T{C}_{*}$. This exact sequence is the well known Künneth’s correlation. In the present paper Künneth’s correlation is generalized. Namely, the conditions are found under which the infinite exact sequence

$\mathrm{\cdots }⟶{T}^{\left(2i+1\right)}{H}_{n+i+1}⟶\mathrm{\cdots }⟶{T}^{\left(3\right)}{H}_{n+2}⟶{T}^{\left(1\right)}{H}_{n+1}⟶{H}_{n}\left(T{C}_{*}\right)$$⟶T{H}_{n}\left({C}_{*}\right)⟶{T}^{\left(2\right)}{H}_{n+1}⟶{T}^{\left(4\right)}{H}_{n+2}⟶\mathrm{\cdots }⟶{T}^{\left(2i\right)}{H}_{n+i}⟶\mathrm{\cdots }$

holds, where ${T}^{\left(2i+1\right)}{H}_{n+i+1}={T}^{\left(2i+1\right)}{H}_{n+i+1}\left({C}_{*}\right)$, ${T}^{\left(2i\right)}{H}_{n+i}={T}^{\left(2i\right)}{H}_{n+i}\left({C}_{*}\right)$. The formula makes it possible to generalize Milnor’s formula for the cohomologies of an arbitrary complex, relatively to the Kolmogorov homology to the Alexandroff–Čech homology for a compact space, to a generative result of Massey for a local compact Hausdorff space X and a direct system $\left\{U\right\}$ of open subsets U of X such that $\overline{U}$ is a compact subset of X.

MSC 2010: 55N10

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Accepted: 2018-12-24

Published Online: 2019-04-09

Published in Print: 2019-06-01

Citation Information: Georgian Mathematical Journal, Volume 26, Issue 2, Pages 295–301, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X,

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